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The hardening soil model: Formulation and verification

T. SchanzLaboratory of Soil Mechanics, Bauhaus-University Weimar, Germany

P.A. VermeerInstitute of Geotechnical Engineering, University Stuttgart, Germany

P.G. BonnierPLAXIS B.V., Netherlands

Keywords: constitutive modeling, HS-model, calibration, verification

ABSTRACT: A new constitutive model is introduced which is formulated in the framework ofclassical theory of plasticity. In the model the total strains are calculated using a stress-dependentstiffness, different for both virgin loading and un-/reloading. The plastic strains are calculated byintroducing a multi-surface yield criterion. Hardening is assumed to be isotropic depending on boththe plastic shear and volumetric strain. For the frictional hardening a non-associated and for the caphardening an associated flow rule is assumed.

First the model is written in its rate form. Therefor the essential equations for the stiffness mod-ules, the yield-, failure- and plastic potential surfaces are given.

In the next part some remarks are given on the models incremental implementation in thePLAXIS computer code. The parameters used in the model are summarized, their physical interpre-tation and determination are explained in detail.

The model is calibrated for a loose sand for which a lot of experimental data is available. Withthe so calibrated model undrained shear tests and pressuremeter tests are back-calculated.

The paper ends with some remarks on the limitations of the model and an outlook on further de-velopments.

1 INTRODUCTION

Due to the considerable expense of soil testing, good quality input data for stress-strain relation-ships tend to be very limited. In many cases of daily geotechnical engineering one has good data onstrength parameters but little or no data on stiffness parameters. In such a situation, it is no help toemploy complex stress-strain models for calculating geotechnical boundary value problems. In-stead of using Hooke's single-stiffness model with linear elasticity in combination with an idealplasticity according to Mohr-Coulomb a new constitutive formulation using a double-stiffnessmodel for elasticity in combination with isotropic strain hardening is presented.

Summarizing the existing double-stiffness models the most dominant type of model is the Cam-Clay model (Hashiguchi 1985, Hashiguchi 1993). To describe the non-linear stress-strain behav-iour of soils, beside the Cam-Clay model the pseudo-elastic (hypo-elastic) type of model has beendeveloped. There an Hookean relationship is assumed between increments of stress and strain andnon-linearity is achieved by means of varying Young's modulus. By far the best known model ofthis category ist the Duncan-Chang model, also known as the hyperbolic model (Duncan & Chang1970). This model captures soil behaviour in a very tractable manner on the basis of only two stiff-ness parameters and is very much appreciated among consulting geotechnical engineers. The majorinconsistency of this type of model which is the reason why it is not accepted by scientists is that,in contrast to the elasto-plastic type of model, a purely hypo-elastic model cannot consistently dis-tinguish between loading and unloading. In addition, the model is not suitable for collapse loadcomputations in the fully plastic range.

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These restrictions will be overcome by formulating a model in an elasto-plastic framework inthis paper. Doing so the Hardening-Soil model, however, supersedes the Duncan-Chang model byfar. Firstly by using the theory of plasticity rather than the theory of elasticity. Secondly by includ-ing soil dilatancy and thirdly by introducing a yield cap.

In contrast to an elastic perfectly-plastic model, the yield surface of the Hardening Soil model isnot fixed in principal stress space, but it can expand due to plastic straining. Distinction is madebetween two main types of hardening, namely shear hardening and compression hardening. Shearhardening is used to model irreversible strains due to primary deviatoric loading. Compressionhardening is used to model irreversible plastic strains due to primary compression in oedometerloading and isotropic loading.

For the sake of convenience, restriction is made in the following sections to triaxial loadingconditions with 2σ ′ = 3σ ′ and 1σ ′ being the effective major compressive stress.

2 CONSTITUTIVE EQUATIONS FOR STANDARD DRAINED TRIAXIAL TEST

A basic idea for the formulation of the Hardening-Soil model is the hyperbolic relationship be-tween the vertical strain ε1, and the deviatoric stress, q, in primary triaxial loading. When subjectedto primary deviatoric loading, soil shows a decreasing stiffness and simultaneously irreversibleplastic strains develop. In the special case of a drained triaxial test, the observed relationship be-tween the axial strain and the deviatoric stress can be well approximated by a hyperbola (Kondner& Zelasko 1963). Standard drained triaxial tests tend to yield curves that can be described by:

The ultimate deviatoric stress, qf, and the quantity qa in Eq. 1 are defined as:

The above relationship for qf is derived from the Mohr-Coulomb failure criterion, which involvesthe strength parameters c and ϕp. As soon as q = qf , the failure criterion is satisfied and perfectlyplastic yielding occurs. The ratio between qf and qa is given by the failure ratio Rf, which shouldobviously be smaller than 1. Rf = 0.9 often is a suitable default setting. This hyperbolic relationshipis plotted in Fig. 1.

2.1 Stiffness for primary loading

The stress strain behaviour for primary loading is highly nonlinear. The parameter E50 is the con-fining stress dependent stiffness modulus for primary loading. E50 is used instead of the initialmodulus Ei for small strain which, as a tangent modulus, is more difficult to determine experimen-tally. It is given by the equation:

refE50 is a reference stiffness modulus corresponding to the reference stress refp . The actual stiff-

ness depends on the minor principal stress, 3σ ′ , which is the effective confining pressure in a tri-axial test. The amount of stress dependency is given by the power m. In order to simulate a loga-rithmic stress dependency, as observed for soft clays, the power should be taken equal to 1.0. As a

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Figure 1. Hyperbolic stress-strain relation in primary loading for a standard drained triaxial test.

secant modulus refE50 is determined from a triaxial stress-strain-curve for a mobilization of 50% ofthe maximum shear strength qf .

2.2 Stiffness for un-/reloading

For unloading and reloading stress paths, another stress-dependent stiffness modulus is used:

where refurE is the reference Young's modulus for unloading and reloading, corresponding to thereference pressure σ ref. Doing so the un-/reloading path is modeled as purely (non-linear) elastic.The elastic components of strain εe are calculated according to a Hookean type of elastic relationusing Eqs. 4 + 5 and a constant value for the un-/reloading Poisson's ratio υur.

For drained triaxial test stress paths with σ2 = σ3 = constant, the elastic Young's modulus Eur re-mains constant and the elastic strains are given by the equations:

Here it should be realised that restriction is made to strains that develop during deviatoric loading,whilst the strains that develop during the very first stage of the test are not considered. For the firststage of isotropic compression (with consolidation), the Hardening-Soil model predicts fully elasticvolume changes according to Hooke's law, but these strains are not included in Eq. 6.

2.3 Yield surface, failure condition, hardening law

For the triaxial case the two yield functions f12 and f13 are defined according to Eqs. 7 and 8. Here

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Figure 2. Successive yield loci for various values of the hardening parameter γ p and failure surface.

the measure of the plastic shear strain γ p according to Eq. 9 is used as the relevant parameter forthe frictional hardening:

with the definition

In reality, plastic volumetric strains pυε will never be precisely equal to zero, but for hard soilsplastic volume changes tend to be small when compared with the axial strain, so that the approxi-mation in Eq. 9 will generally be accurate.

For a given constant value of the hardening parameter, γ p, the yield condition f12 = f13 = 0 can bevisualised in p'-q-plane by means of a yield locus. When plotting such yield loci, one has to useEqs. 7 and 8 as well as Eqs. 3 and 4 for E50 and Eur respectively. Because of the latter expressions,the shape of the yield loci depends on the exponent m. For m = 1.0 straight lines are obtained, butslightly curved yield loci correspond to lower values of the exponent. Fig. 2 shows the shape ofsuccessive yield loci for m = 0.5, being typical for hard soils. For increasing loading the failure sur-faces approach the linear failure condition according to Eq. 2.

2.4 Flow rule, plastic potential functions

Having presented a relationship for the plastic shear strain, γ p, attention is now focused on theplastic volumetric strain pυε . As for all plasticity models, the Hardening-Soil model involves a re-lationship between rates of plastic strain, i.e. a relationship between pυε� and pγ� . This flow rule hasthe linear form:

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Clearly, further detail is needed by specifying the mobilized dilatancy angle mψ . For the presentmodel, the expression:

is adopted, where cvϕ is the critical state friction angle, being a material constant independent ofdensity (Schanz & Vermeer 1996), and mϕ is the mobilized friction angle:

The above equations correspond to the well-known stress-dilatancy theory (Rowe 1962, Rowe1971), as explained by (Schanz & Vermeer 1996). The essential property of the stress-dilatancytheory is that the material contracts for small stress ratios mϕ < cvϕ , whilst dilatancy occurs forhigh stress ratiosmϕ < cvϕ . At failure, when the mobilized friction angle equals the failure angle,

pϕ , it is found from Eq. 11 that:

Hence, the critical state angle can be computed from the failure angles pϕ and pψ . The above defi-nition of the flow rule is equivalent to the definition of definition of the plastic potential functionsg12 and g13 according to:

Using the Koiter-rule (Koiter 1960) for yielding depending on two yield surfaces (Multi-surfaceplasticity) one finds:

Calculating the different plastic strain rates by this equation, Eq. 10 directly follows.

3 TIME INTEGRATION

The model as described above has been implemented in the finite element code PLAXIS (Vermeer& Brinkgreve 1998). To do so, the model equations have to be written in incremental form. Due tothis incremental formulation several assumptions and modifications have to be made, which will beexplained in this section.

During the global iteration process, the displacement increment follows from subsequent solu-tion of the global system of equations:

where K is the global stiffness matrix in which we use the elastic Hooke's matrix D, fext is a globalload vector following from the external loads and f int is the global reaction vector following fromthe stresses. The stress at the end of an increment σ 1 can be calculated (for a given strain increment∆ε) as:

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whereσ0 , stress at the start of the increment,

∆σ , resulting stress increment,4D , Hooke's elasticity matrix, based on the unloading-reloading stiffness,

∆ε , strain increment (= B∆u),

γ p , measure of the plastic shear strain, used as hardening parameter,

∆Λ , increment of the non-negative multiplier,

g , plastic potential function.

The multiplier Λ has to be determined from the condition that the function f (σ1, γ p) = 0 has to bezero for the new stress and deformation state.

As during the increment of strain the stresses change, the stress dependant variables, like theelasticity matrix and the plastic potential function g, also change. The change in the stiffness duringthe increment is not very important as in many cases the deformations are dominated by plasticity.

This is also the reason why a Hooke's matrix is used. We use the stiffness matrix 4D based on the

stresses at the beginning of the step (Euler explicit). In cases where the stress increment followsfrom elasticity alone, such as in unloading or reloading, we iterate on the average stiffness duringthe increment.

The plastic potential function g also depends on the stresses and the mobilized dilation anglemψ . The dilation angle for these derivatives is taken at the beginning of the step. The implementa-

tion uses an implicit scheme for the derivatives of the plastic potential function g. The derivativesare taken at a predictor stress σtr, following from elasticity and the plastic deformation in the previ-ous iteration:

The calculation of the stress increment can be performed in principal stress space. Therefore ini-tially the principal stresses and principal directions have to be calculated from the Cartesianstresses, based on the elastic prediction. To indicate this we use the subscripts 1, 2 and 3 and have

321 σσσ ≥≥ where compression is assumed to be positive.Principal plastic strain increments are now calculated and finally the Cartesian stresses have to

be back-calculated from the resulting principal constitutive stresses. The calculation of the consti-tutive stresses can be written as:

From this the deviatoric stress q (σ1 – σ3) and the asymptotic deviatoric stress qa can be expressedin the elastic prediction stresses and the multiplier ∆Λ:

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where

For these stresses the function

should be zero. As the increment of the plastic shear strain ∆γ p also depends linearly on the multi-plier ∆Λ, the above formulae result in a (complicated) quadratic equation for the multiplier ∆Λwhich can be solved easily. Using the resulting value of ∆Λ, one can calculate (incremental)stresses and the (increment of the) plastic shear strain.

In the above formulation it is assumed that there is a single yield function. In case of triaxialcompression or triaxial extension states of stress there are two yield functions and two plastic po-tential functions. Following (Koiter 1960) one can write:

where the subscripts indicate the principal stresses used for the yield and potential functions. Atmost two of the multipliers are positive. In case of triaxial compression we have σ2 = σ3, Λ23 = 0and we use two consistency conditions instead of one as above. The increment of the plastic shearstrain has to be expressed in the multipliers. This again results in a quadratic equation in one of themultipliers.

When the stresses are calculated one still has to check if the stress state violates the yield crite-rion q ≤ qf. When this happens the stresses have to be returned to the Mohr-Coulomb yield surface.

4 ON THE CAP YIELD SURFACE

Shear yield surfaces as indicated in Fig. 2 do not explain the plastic volume strain that is measuredin isotropic compression. A second type of yield surface must therefore be introduced to close theelastic region in the direction of the p-axis. Without such a cap type yield surface it would not bepossible to formulate a model with independent input of both E50 and Eoed. The triaxial moduluslargely controls the shear yield surface and the oedometer modulus controls the cap yield surface.In fact, refE50 largely controls the magnitude of the plastic strains that are associated with the shearyield surface. Similarly, refoedE is used to control the magnitude of plastic strains that originate fromthe yield cap. In this section the yield cap will be described in full detail. To this end we considerthe definition of the cap yield surface (a = c cot ϕ):

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where M is an auxiliary model parameter that relates to NCK 0 as will be discussed later. Furthermore we have p = (σ1 + σ2 + σ3) and

with

q is a special stress measure for deviatoric stresses. In the special case of triaxial compression ityields q = (σ1 – σ3) and for triaxial extension reduces to q = α (σ1 – σ3). For yielding on the capsurface we use an associated flow rule with the definition of the plastic potential gc:

The magnitude of the yield cap is determined by the isotropic pre-consolidation stress pc. For thecase of isotropic compression the evolution of pc can be related to the plastic volumetric strain rate

pvε� :

Here H is the hardening modulus according to Eq. 32, which expresses the relation between theelastic swelling modulus Ks and the elasto-plastic compression modulus Kc for isotropic compres-sion:

From this definition follows a stress dependency of H. For the case of isotropic compression wehave q = 0 and therefor cpp �� = . For this reason we find Eq. 33 directly from Eq. 31:

The plastic multiplier c� referring to the cap is determined according to Eq. 35 using the addi-tional consistency condition:

Using Eqs. 33 and 35 we find the hardening law relating pc to the volumetric cap strain cvε :

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Figure 3. Representation of total yield contour of the Hardening-Soil model in principal stress space for co-hesionless soil.

The volumetric cap strain is the plastic volumetric strain in isotropic compression. In addition tothe well known constants m and σref there is another model constant H. Both H and M are cap pa-rameters, but they are not used as direct input parameters. Instead, we have relationships of the

form NCK0 = NCK0 (..., M, H) and refoedE =

refoedE (..., M, H), such that NCK0 and

refoedE can be used as in-

put parameters that determine the magnitude of M and H respectively. The shape of the yield cap isan ellipse in p – q~ -plane. This ellipse has length pc + a on the p-axis and M (pc + a) on the q~ -axis.Hence, pc determines its magnitude and M its aspect ratio. High values of M lead to steep caps un-derneath the Mohr-Coulomb line, whereas small M-values define caps that are much more pointedaround the p-axis.

For understanding the yield surfaces in full detail, one should consider Fig. 3 which depictsyield surfaces in principal stress space. Both the shear locus and the yield cap have the hexagonalshape of the classical Mohr-Coulomb failure criterion. In fact, the shear yield locus can expand upto the ultimate Mohr-Coulomb failure surface. The cap yield surface expands as a function of thepre-consolidation stress pc.

5 PARAMETERS OF THE HARDENING-SOIL MODEL

Some parameters of the present hardening model coincide with those of the classical non-hardeningMohr-Coulomb model. These are the failure parameters ϕp,, c and ψp. Additionally we use the ba-sic parameters for the soil stiffness:

refE50 , secant stiffness in standard drained triaxial test,

refoedE , tangent stiffness for primary oedometer loading and

m, power for stress-level dependency of stiffness.

This set of parameters is completed by the following advanced parameters:

refurE , unloading/ reloading stiffness,

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vur , Poisson's ratio for unloading-reloading,

pref, reference stress for stiffnesses,

NCK0 , K0-value for normal consolidation and

Rf, failure ratio qf / qa.

Experimental data on m, E50 and Eoed for granular soils is given in (Schanz & Vermeer 1998).

5.1 Basic parameters for stiffness

The advantage of the Hardening-Soil model over the Mohr-Coulomb model is not only the use of ahyperbolic stress-strain curve instead of a bi-linear curve, but also the control of stress level de-pendency. For real soils the different modules of stiffness depends on the stress level. With theHardening-Soil model a stiffness modulusrefE50 is defined for a reference minor principal stress ofσ3 = σref. As some readers are familiar with the input of shear modules rather than the above stiff-ness modules, shear modules will now be discussed. Within Hooke's theory of elasticity conversionbetween E and G goes by the equation E = 2 (1 + v) G. As Eur is a real elastic stiffness, one maythus write Eur = 2 (1 + vur) Gur, where Gur is an elastic shear modulus. In contrast to Eur, the secantmodulus E50 is not used within a concept of elasticity. As a consequence, there is no simple conver-sion from E50 to G50. In contrast to elasticity based models, the elasto-plastic Hardening-Soil modeldoes not involve a fixed relationship between the (drained) triaxial stiffness E50 and the oedometerstiffness Eoed. Instead, these stiffnesses must be given independently. To define the oedometer stiff-ness we use

where Eoed is a tangent stiffness modulus for primary loading. Hence, refoedE is a tangent stiffness ata vertical stress of σ1 = σref.

5.2 Advanced parameters

Realistic values of vur are about 0.2. In contrast to the Mohr-Coulomb model, NCK0 is not simply afunction of Poisson's ratio, but a proper input parameter. As a default setting one can use the highlyrealistic correlation NCK0 = 1 – sin ϕp. However, one has the possibility to select different values.All possible different input values for NCK0 cannot be accommodated for. Depending on other pa-rameters, such as E50, Eoed, Eur and vur, there happens to be a lower bound on NCK0 . The reason forthis situation will be explained in the next section.

5.3 Dilatancy cut-off

After extensive shearing, dilating materials arrive in a state of critical density where dilatancy hascome to an end. This phenomenon of soil behaviour is included in the Hardening-Soil model bymeans of a dilatancy cut-off. In order to specify this behaviour, the initial void ratio, e0, and themaximum void ratio, ecv, of the material are entered. As soon as the volume change results in astate of maximum void, the mobilized dilatancy angle, ψm, is automatically set back to zero, as in-dicated in Eq. 38 and Fig. 4:

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Figure 4. Resulting strain curve for a standard drained triaxial test including dilatancy cut-off.

The void ratio is related to the volumetric strain, εv by the relationship:

where an increment of εv is negative for dilatancy. The initial void ratio, e0, is the in-situ void ratioof the soil body. The maximum void ratio, ecv, is the void ratio of the material in a state of criticalvoid (critical state).

6 CALIBRATION OF THE MODEL

In a first step the Hardening-Soil model was calibrated for a sand by back-calculating both triaxialcompression and oedometer tests. Parameters for the loosely packed Hostun-sand (e0 = 0.89), awell known granular soil in geotechnical research, are given in Tab. 1. Figs. 5 and 6 show the satis-fying comparison between the experimental (three different tests) and the numerical result. For theoedometer tests the numerical results consider the unloading loop at the maximum vertical loadonly.

7 VERIFICATION OF THE MODEL

7.1 Undrained behaviour of loose Hostun-sand

In order to verify the model in a first step two different triaxial compression tests on loose Hostun-sand under undrained conditions (Djedid 1986) were simulated using the identical parameter fromthe former calibration. The results of this comparison are displayed in Figs. 7 and 8.

In Fig. 7 we can see that for two different confining pressures of σc = 300 and 600 kPa the stresspaths in p-q-space coincide very well. For deviatoric loads of q ≈ 300 kPa excess porewater pres-sures tend to be overestimated by the calculations.

Additionally in Fig. 8 the stress-strain-behaviour is compared in more detail. This diagram con-tains two different sets of curves. The first set (•, ♠) relates to the axial strain ε1 at the horizontal

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Figure 5. Comparison between the numerical (•) and experimental results for the oedometer tests.

Figure 6. Comparison between the numerical (•) and experimental results for the drained triaxial tests(σ3 = 300 kPa) on loose Hostun-sand.

and the effective stress ratio 31 / σσ ′′ on the vertical (left) axis. The second set (o, ) refers to thenormalised excess pore water pressure ∆u/σc on the right vertical axis. Experimental results forboth confining stresses are marked by symbols, numerical results by straight and dotted lines.

Analysing the amount of effective shear strength it can be seen that the maximum calculatedstress ratio falls inside the range of values from the experiments. The variation of effective frictionfrom both tests is from 33.8 to 35.4 degrees compared to an input value of 34 degrees. Axial stiff-ness for a range of vertical strain of ε1 < 0.05 seems to be slightly over-predicted by the model. Dif-ferences become more pronounced for the comparison of excess pore water pressure generation.

Here the calculated maximum amount of ∆u is higher then the measured values. The rate of de-crease in ∆u for larger vertical strain falls in the range of the experimental data.

Table 1. Parameters of loose Hostun-sand.

vur m ϕp ψp refrefs EE 50/refref

ur EE 50/refE50

0.20 0.65 34° 0° 0.8 3.0 20 MPa

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Figure 7. Undrained behaviour of loose Hostun-sand: p-q-plane.

Figure 8. Undrained behaviour of loose Hostun-sand: stress-strain relations.

7.2 Pressuremeter test Grenoble

The second example to verify the Hardening-Soil model is a back-calculation of a pressuremetertest on loose Hostun-sand. This test is part of an experimental study using the calibration chamberat the IMG in Grenoble (Branque 1997). This experimental testing facility is shown in Fig. 9.

The cylindrical calibration chamber has a height of 150 cm and a diameter of 120 cm. In thetest considered in the following a vertical surcharge of 500 kPa is applied at the top of the soil massby a membrane. Because of the radial deformation constraint the state of stress can be interpretedin this phase as under oedometer conditions. Inside the chamber a pressuremeter sonde of a radiusr0 of 2.75 cm and a length of 16 cm is placed. For the test considered in the following examplethere was loose Hostun-sand (Dr ≈ 0.5) of a density according to the material parameters as shownin Tab. 1 placed around the pressuremeter by pluviation. After the installation of the device and thefilling of the chamber the pressure is increased and the resulting volume change is registered.

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Figure 9. Pressuremeter Grenoble.

This experimental setup was modeled within a FE-simulation as shown in Fig. 10. On the lefthand side the axis-symmetric mesh and its boundary conditions is displayed. The dimensions arethose of the complete calibration chamber. In the left bottom corner of the geometry the mesh isfiner because there the pressuremeter is modeled.

In the first calculation phase the vertical surcharge load A is applied. At the same time the hori-zontal load B is increased the way practically no deformations occur at the free deformation bound-ary in the left bottom corner. In the second phase the load group A is kept constant and the loadgroup B is increased according to the loading history in the experiment. The (horizontal) deforma-tions are analysed over the total height of the free boundary. In order to (partly) get rid of the de-formation constrains at the top of this boundary, marked point A in the detail on the right hand sideof Fig. 10 two interfaces were placed crossing each other in point A. Fig. 11 shows the comparisonof the experimental and numerical results for the test with a vertical surcharge of 500 kPa.

On the vertical axis the pressure (relating to load group B) is given and on the horizontal axisthe volumetric deformation of the pressuremeter. Because the calculation was run taking into ac-count large deformations (updated mesh analysis) the pressure p in the pressuremeter has to be cal-culated from load multiplier ΣLoad B according to Eq. 40, taking into account the mean radial de-formation ∆r of the free boundary:

The agreement between the experimental and the numerical data is very good, both for the initialpart of phase 2 and for larger deformations of up to 30%.

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Figure 10. FE-discretization of the calibration chamber as used in the calculation.

Figure 11. Comparison between experimental and numerical results of the pressuremeter test.

8 CONCLUDING REMARKS

A new constitutive model was introduced which is formulated in the framework of hardeningmulti-surface plasticity. The model was described in the essential equations as the ones for theyield- and plastic potential surfaces and the applied hardening laws.

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After implementation of the model in the PLAXIS code it was calibrated for a loose sand usingboth oedometer and drained triaxial test data. With this unique set of parameters undrained sheartests and a pressuremeter test run in a calibration chamber were numerically simulated. For bothverifications of the model the comparison between experimental and numerical is very promising:main characteristics of hardening soil behaviour can be described both in a qualitative and quanti-tative way. Because of the used set of input parameters, all parameters have a clear geotechnicalrelevance, the model is very attractive for the use in daily geotechnical practise.

Further developments include an extensions of database in the field of stiffness parameters forcohesive soils, the application of the model for boundary value problems in which a small strainstiffness is relevant and the 3-dimensional verification of the model.

REFERENCES

Branque, D. (1997), Utilisation d'un modèle Élasto-plastique avec dilatance dans I'interprétation de l'essaipressiométrique sur sable, PhD thesis, Ecole Centrale Paris.

Djedid, A. (1986), Etude du comportement non-drainé du sable, Mémoire de D.E.A., Institut de Mécaniquede Grenoble.

Duncan, J.M. & Chang, C.Y (1970), Nonlinear analysis of stress and strain in soil, J. Soil Mech. Found. Div.ASCE 96, 1629-1653.

Hashiguchi, K. (1985), Macrometric approaches-static-instrinsically time-independent, in Proc. XI ICSMFE,San Francisco, Constitutive laws for soils, pp. 25-65.

Hashiguchi, K. (1993), Fundamental requirements and formulations of elastoplastic constitutive equationswith tangential plasticity, Int. J. Plasticity 9, 525-549.

Koiter, W.T. (1960), General theorems for elastic-plastic solids, in Sneddon & Hill, eds, Progress in SolidMechanics, North Holland Publishing Co., Amsterdam, pp. 165-221.

Kondner, R.L. & Zelasko, J.S. (1963), A hyperbolic stress strain formulation for sands, Proc. 2nd Pan. Am.ICOSFE Brazil 1, 289-394.

Rowe, P.W. (1962), The stress-dilatancy relation for static equilibrium of an assembly of particles in contact,Proc. Roy. Soc. A. 269, 500-527.

Rowe, P.W. (1971), Theoretical meaning and observed values of deformation parameters for soil, in Proc. ofRoscoe memorial symposium, Foulis, Henley-on-Thames, pp. 143-194.

Schanz, T. & Vermeer, P.A. (1996), Angles of friction and dilatancy of sand, Géotechnique 46, No. 1, 145-151.

Schanz, T. & Vermeer, P.A. (1998), On the stiffness of sands, Géotechnique 48, 383-387.Vermeer, P.A. & Brinkgreve, R.B.J. (1998), PLAXIS: Finite element code for soil and rock analyses (Version

7.l), B alkema, Rotterdam.